Kalman Filtering with Intermittent Observations: Weak Convergence to a Stationary Distribution

The paper studies the asymptotic behavior of discrete time Random Riccati Equations (RRE) arising in Kalman filtering when the arrival of the observations is described by a Bernoulli independent and identically distributed (i.i.d.) process. We model the RRE as an order-preserving, strongly sublinear random dynamical system (RDS). Under a sufficient condition, stochastic boundedness, and using a limit-set dichotomy result for order-preserving, strongly sublinear RDS, we establish the asymptotic properties of the RRE: the sequence of random prediction error covariance matrices converges weakly to a unique invariant distribution, whose support exhibits fractal behavior. For stabilizable and detectable systems, stochastic boundedness (and hence weak convergence) holds for any nonzero observation packet arrival probability and, in particular, we can establish weak convergence at operating arrival rates well below the critical probability for mean stability (the resulting invariant measure in that situation does not possess a first moment). We apply the weak-Feller property of the Markov process governing the RRE to characterize the support of the limiting invariant distribution as the topological closure of a countable set of points, which, in general, is not dense in the set of positive semi-definite matrices. We use the explicit characterization of the support of the invariant distribution and the almost sure (a.s.) ergodicity of the sample paths to easily compute statistics of the invariant distribution. A one-dimensional example illustrates that the support is a fractured subset of the non-negative reals with self-similarity properties.